I. Field of the Invention
The present invention relates to CDMA (Code Division Multiple Access) cellular telephone and wireless data communications with data rates up to multiple T1 (1.544 Mbps) and higher (>100 Mbps), and to optical CDMA. Applications are mobile, point-to-point and satellite communication networks. More specifically the present invention relates to a new and novel approach to the design of waveforms and filters using mathematical formulations which generalize the Wavelet concept to communications and radar.
II. Description of the Related Art
Multi-resolution waveforms used for signaling and/or filters and which are addressed in this invention are defined to be waveforms of finite extent in time and frequency, with scale and shift properties of multi-resolution over the time-frequency (t−f) space. These waveforms can also be referred to as multi-scale waveforms, and include multi-rate filtering and the Wavelet as special cases. The emphasis will be on digital design and applications with the understanding that these multi-resolution waveforms are equally applicable to analog design and applications.
Background art consists of the collection of waveform and filtering design techniques which can be grouped into six broad categories. These categories are: C1) least squares (LS) design algorithms for filters and waveforms that design to specifications on their frequency response, C2) analytic filters and waveforms which are specified by a few free design parameters that can be sub-categorized into current applications and theorectical studies, C3) combinations of C1 and C2 for greater flexibility in meeting communications and radar performance goals, C4) special design techniques to control the noise levels from intersymbol interference (ISI) and adjacent channel interference (ACI) in the presence of timing offsets for multiple channel applications, C5) Wavelet filter design using scaling functions (iterated filter banks) as the set of design coordinates or basis functions, C6) filter and waveform design techniques for non-linear channels and in particular for operation in the non-linear and saturation regions of a high power amplifier (HPA) such as a traveling wave tube (TWT) or a solid state amplifier, and C7) LS dynamic filters derived from discrete filtering and tracking algorithms that include adaptive equalization for communications, adaptive antenna filters, Wiener filters, Kalman filters, and stochastic optimization filters.
Category C1 common examples of LS digital filter design are the eigenvalue algorithm in “A New Approach to Least-Squares FIR Filter Design and Applications Including Nyquist Filters” and the Remez-Exchange algorithm in “A Computer Program for Designing Optimum FIR Linear Phase Filters’. The eigenvalue algorithm is a direct LS minimization and the Remez-Exchange can be reformulated as an equivalent LS gradient problem through proper choice of the cost function. Both LS algorithms use the FIR (finite impulse response) digital samples as the set of design coordinates. LS design metrics are the error residuals in meeting their passband and stopband ideal performance as shown in FIG. 3. Category C2 common examples of analytical waveforms and filters for system applications are the analog Chebyshev, Elliptic, Butterworth, and the digital raised-cosine, and square-root raised-cosine. For theorectical studies, common examples are polyphase multirate filters, quadrature mirror filters (QMF), and perfect reconstruction filters. Although these theorectical studies have yet to yield realizable useful filters for system applications, their importance for this invention lies in their identification and application of ideal performance metrics for filter designs. Category C3 common digital example is to start with the derivation of a Remez-Exchange FIR filter and then up-sample and filter with another bandwidth limiting filter. This results in an FIR over the desired frequency band that is larger than available with the Remez-Exchange algorithm and with sidelobes that now drop off with frequency compared to the flat sidelobes of the original Remez-Exchange FIR. A category C4 common example is to select the free parameters of the category C3 filter in order to minimize the signal to noise power ratio of the data symbol (SNR) losses from the ISI, ACI, and the non-ideal demodulation. A second common example is start with a truncated pulse whose length is short enough compared to the symbol repetition interval, to accommodate the timing offsets without significant impact on the ISI and ACI SNR losses. This shortened pulse can then be shaped in the frequency domain.
Category C5 Wavelet filter design techniques discussed in the next section will serve as a useful reference in the disclosure of this invention. Category C6 common example is the Gaussian minimum shift keying (GMSK) waveform. This is a constant amplitude phase encoded Gaussian waveform which has no sidelobe re-growth through a non-linear or saturating HPA. Category C7 common examples are the adaptive equalization filter for communication channels, the adaptive antenna filter, and the Kalman filter for applications including target tracking and prediction as well as for equalization and adaptive antennas.
Minimizing excess bandwidth in the waveform and filter design is a key goal in the application of the C1, . . . , C6 design techniques for communications and radar. Excess bandwidth is identified as the symbol α in the bandwidth-time product BTs=1+α where the two-sided available frequency band is B and the symbol repetition interval is Ts. Current performance capability is represented by the use of the square-root raised-cosine (sq-rt r-c) waveform with α=0.22 to 0.4 as shown in FIG. 8. The goal is to design a waveform with α=0 within the performance constraints of ISI, ACI, passband, sideband, and passband ripple. This goal of eliminating the excess bandwidth corresponds to the symbol rate equal to the available frequency band 1/Tx=B/(1+α)=B for α=0. This symbol rate 1/Ts=B is well known to be the maximum possible rate for which orthogonality between symbols is maintained. A fundamental performance characteristic of the new waveform designs is the ability to eliminate excess bandwidth for many applications.
Scope of this invention will include all of the waveform and filter categories with the exception of category C4 special design techniques and category C7 LS dynamic filters. Emphasis will be on the category C5 Wavelets to establish the background art since Wavelets are multi-resolution waveforms that can eliminate the excess bandwidth and have known design algorithms for FIR waveforms and filters. However, they do not have a design mechanism that allows direct control of the ISI, ACI, passband, sideband, and passband ripple. Category C2 theorectical studies also eliminate the excess bandwidth. However, they are not multi-resolution waveforms and do not have realizable FIR design algorithms. So the emphasis in background art will be on Wavelets whose relevant properties we briefly review.
Wavelet background art relevant to this invention consists of the discrete Waveform equations and basic properties, application of Wavelets to cover a discrete digital time-frequency (t−f) signal space, and the design of Wavelets using the iterated filter construction. Wavelets are waveforms of finite extent in time (t) and frequency (f) over the t−f space, with multi-resolution, scaling, and translation properties. Wavelets over the analog and digital t−f spaces respectively are defined by equations (1) and (2) as per Daubechies's “Ten Lectures on Wavelets”, Philadelphia:_SIAM, 1992
Continuous Wavelet
                                                        ψ              ⁢                                                                                  a              ,              b                                ⁢                      (            t            )                          =                                                          a                                                    -                              1/2                                              ⁢                      ψ            ⁡                          (                                                t                  -                  b                                a                            )                                                          (        1        )            
Discrete Wavelet
                                                        ψ              ⁢                                                                                  a              ,              b                                ⁢                      (            n            )                          =                                                          a                                                    -                              1/2                                              ⁢                      ψ            ⁡                          (                                                n                  -                  b                                a                            )                                                          (        2        )            where the two index parameters “a,b” are the Wavelet dilation and translation respectively or equivalently are the scale and shift. The ψ is the “mother” wavelet and is a real and symmetric localized function in the t−f space used to generate the doubly indexed Wavelet ψab. The scale factor “|α|−1/2”, has been chosen to keep the norm of the Wavelet invariant under the parameter change “a,b”. Norm is the square root of the energy of the Wavelet response. The Wavelets ψa,b and ψ are localized functions in the t−f space which means that both their time and frequency lengths are bounded. The discrete Wavelet has the time “t” replaced by the equivalent digital sample number “n” assuming the waveform is uniformly sampled at “T” second intervals.
Wavelets in digital t−f space have an orthogonal basis that is obtained by restricting the choice of the parameters “a,b” to the values a=2−P, b=qM2P where “p, q” are the new scale and translation parameters and “M” is the spacing or repetition interval Ts=MT of the Wavelets (which from a communications viewpoint are symbols) at the same scale “p”. Wavelets at “p,q” are related to the mother Wavelet by the equationψp,q(n)=2−p/2ψ(2−pn−qM)  (3)where the mother Wavelet is a real and even function of the sample coordinates at dc (dc refers to the origin f=0 of the frequency f space). The orthonormality property means that these Wavelets satisfy the orthogonality equation with a correlation value equal to “1”.
                                                                                          ∑                  n                                                                                        ⁢                                                                  ⁢                                                      ψ                                          p                      ,                      q                                                        ⁢                                      ψ                                          k                      ,                      m                                                                                  =                                                1                  ⁢                                                                          ⁢                  if                  ⁢                                                                          ⁢                  both                  ⁢                                                                          ⁢                  p                                =                                                      k                    ⁢                                                                                  ⁢                    and                    ⁢                                                                                  ⁢                    q                                    =                  m                                                                                                        =                              0                ⁢                                                                  ⁢                otherwise                                                                        (        4        )            
Wavelet representation of a digital t−f space starts with selecting an N sample time window of a uniform stream of digital samples at the rate of 1/T Hz (1/second) equivalent to a “T” second sampling interval. The N point or sample t−f space in FIG. 1 illustrates a Wavelet representation or “tiling” with Wavelets that are designed analytically or by an iterated filter construction.
The t−f space in FIG. 1 is partitioned or covered or tiled by a set of Wavelet subspaces {Wp,p=0, 1, . . . , M−1} where N=2m. Each Wavelet subspace Wp 1 at scale “p” consists of the set of Wavelet time translations {q=0, 1, . . . , N/2p+1−1l} over this subspace. These Wavelet subspaces are mutually orthogonal and the Wavelets within each subspace are mutually orthogonal with respect to the time translates. This N-point t−f space extends over the time interval from 0 to (N−1)T 2 and over the frequency interval from 0 to (N−1) 3 in units of the normalized frequency fNT 4.
The iterated filter bank in FIG. 2 is used to generate the Wavelets which cover the t−f space in FIG. 1. Each filter stage 5 consists of a high pass filter (HPF) and a low pass filter (LPF). Output 6 of the LPF is subsampled by 2 which is equivalent to decimation by 2. This t−f space space is an N-dimensional complex vector metric space V 7. At stage m in the iterated filter bank, the remaining t−f space Vm−1 8 is partitioned into Vm+1 9 and the Wavelet subspace Wm+1 10.
Scaling functions and Wavelets at each stage of this filter bank satisfy the following equations
                                          φ            ⁡                          (              n              )                                =                                    2                                                -                  1                                /                2                                      ⁢                                          ∑                q                            ⁢                                                          ⁢                                                h                  q                                ⁢                                  φ                  ⁡                                      (                                                                  2                        ⁢                        n                                            -                      q                                        )                                                  ⁢                                  ∀                  q                                                                    ⁢                                  ⁢                              ψ            ⁡                          (              n              )                                =                                    2                                                -                  1                                /                2                                      ⁢                                          ∑                q                            ⁢                                                          ⁢                                                g                  q                                ⁢                                  φ                  ⁡                                      (                                                                  2                        ⁢                        n                                            -                      q                                        )                                                  ⁢                                  ∀                  q                                                                                        (        5        )            where σ is the scaling function, ψ is the Wavelet, HPFP coefficients are {hq, ∀q}, LPFp coefficients are {gq, ∀q}, and the equations apply to the stages 0, 1, . . . , m−1. Identifying the scale parameter and using the previous Wavelet formulations enable these equations to be rewritten for stages p=0, 1, . . . , m−1 as
                                          φ            p                    =                                    ∑              q                        ⁢                                                  ⁢                                          h                q                            ⁢                              φ                                                      p                    -                    1                                    ,                  q                                            ⁢                              ∀                p                                                    ⁢                                  ⁢                              ψ            p                    =                                    ∑              q                        ⁢                                                  ⁢                                          g                q                            ⁢                              φ                                                      p                    -                    1                                    ,                  q                                            ⁢                              ∀                p                                                                        (        6        )            For the application the HPFp and LPFp are quadrature mirror filters (QMF) with perfect reconstruction. This means they cover the subspace Vp with flat responses over the subband frequency including the edges of the frequency subband, and the HPFp coefficients are the frequency translated coefficients for the LPFp: {gq=(−1)qhq, ∀q}.
Wavelet design using iterated filter bank starts with the selection of the scaling functions. Starting with a primative scaling function such as the one proposed by Daubechies, one can use the iterated filter construction given by equations (5) and (6) to derive successive approximations to a desired scaling function which has properties that have been designed into it by the selection of the filter coefficients {gq,∀q} at each level of iteration. The Wavelets can be derived from these scaling functions using the iterated filter construction or scaling equations (5) and (6).
Another use of the iterated filter construction is to design the scaling functions as Wavelets thereupon ending up with a larger set of Wavelets for multi-resolution analysis and synthesis as illustrated by Coifman's Wavelets in “Wavelet analysis and signal processing”.